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exp(t aι a)(d dR+d W)exp(t aι a) =d dR+d W+[d dR+d W,t aι a]+12[[d dR+d W,t aι a],t aι a] =d dR+d W =+[d W,t a]12f bc at bt c+r aι at a[d dR+d W,ι a]= a =+12[[d dR+d W,t aι a]12f bc at bt cι a+r aι at a a,t aι a] \begin{aligned} \exp\big( -t^a \wedge \iota_a \big) \big( d_{dR} + d_W \big) \exp\big( t^a \wedge \iota_a \big) & = d_{dR} + d_W + \big[ d_{dR} + d_W, t^a \wedge \iota_a \big] + \tfrac{1}{2} \Big[ \big[ d_{dR} + d_W, t^a \wedge \iota_a \big], t^a \wedge \iota_a \Big] \\ & = d_{dR} + d_W \\ & \phantom{=} + \underset{ - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c + r^a }{ \underbrace{ \big[ d_W , t^a \big] } } \wedge \iota_a - t^a \wedge \underset{ = \mathcal{L}_{a} }{ \underbrace{ \big[ d_{dR} + d_W, \iota_a \big] } } \\ & \phantom{=} + \tfrac{1}{2} \Big[ \underset{ \mathclap{ - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c \iota_a + r^a \iota_a - t^a \mathcal{L}_a } }{ \underbrace{ \big[ d_{dR} + d_W, t^a \wedge \iota_a \big] } } , t^a \wedge \iota_a \Big] \end{aligned}

(wait…)

a circle-principal bundle?

An Ehresmann connection

θ 5Ω 1(Σ 6) \theta^5 \;\in\; \Omega^1\big( \Sigma^6 \big)
HΩ 3(Σ 6) H \;\in\; \Omega^3\big(\Sigma^6\big)
H=dB H = d B
H= 6H H \;=\; \star_6 H
H˜ 5H 5(Hdx 5) \tilde H \coloneqq \star_5 H \coloneqq \star_5 \big( H - \mathcal{F} \wedge d x^5 \big)
Aι 5B A \;\coloneqq\; \iota_5 B
B=Adx 5+B bas B \;=\; A \wedge d x^5 + B^{\mathrm{bas}}
Fd 5A F \coloneqq d_5 A
ι 5H =ι 5dB =dι 5B+[ι 5,d]B =dA+ 5B =F 5Adx 5+ 5(Adx 5+B basic) =F+ 5B bas \begin{aligned} \mathcal{F} &\coloneqq \iota_5 H \\ & = \iota_5 d B \\ & = -d \iota_5 B + [\iota_5, d] B \\ & = - d A + \mathcal{L}_5 B \\ & = - F - \mathcal{L}_5 A \wedge d x^5 + \mathcal{L}_5 ( A \wedge d x^5 + B^{basic} ) \\ & = - F + \mathcal{L}_5 B^{bas} \end{aligned}
6H= 6(dx 5+(Hdx 5))= 5+H˜dx 5 \star_6 H = \star_6 \big( \mathcal{F} \wedge d x^5 + (H - \mathcal{F} \wedge d x^5) \big) = \star_5 \mathcal{F} + \tilde H \wedge d x^5

Last revised on June 25, 2019 at 13:55:28. See the history of this page for a list of all contributions to it.